projective homogeneous $G$-variety is equivariantly isomorphic to a partial flag variety $G/P$ where $P$ is projective variety.

Question

I am looking for a proof(or refference) for this fact

A projective homogeneous $G$-variety is equivariantly isomorphic to a partial flag variety $G/P$ where $P$ is parabolic subgroup.

Answer 1

If $X$ is a projective and homogeneous $G$-variety, then $X=G.x$ for any $x\in X$, in particular $X=G/G_x$. Since $X$ is projective, $G_x$ is parabolic, this is often the definition of being parabolic. See 28.1.3 and 28.1.4 in the Book by Tauvel and Yu, Lie Algebras and Algebraic Groups - the first of these two items also explains why being parabolic is equivalent to fixing a partial flag.